Equation and graph of a hyperbola

Hyperbola in mathematics refers to a smooth curve that lies over a simple plane and defined in terms of different geometric properties. The standard definition of hyperbola following which standard equation and graph has been derived is that defines hyperbola as a locus of all points in a way that difference in distances between any two focal point remains constant.

The equation of a hyperbola with focal point lying on the x axis is given as follows

Figure 1: Diagram of a hyperbola

In the above figure of hyperbola, the origin lies at the center. The respective coordinates of two focal points of the hyperbola are (-c,0) and (c,0).  All the points located at same distances of these points are set in the diagram. Center of the hyperbola is given as (0,0). Suppose a point P, having value +a is taken on the hyperbola. The difference between P and the two focal point then can be given as

Now, if any point (x, y) chosen on the hyperbola, then an equation can be obtained using the distant formula

Different part of a hyperbola

The standard form a hyperbola where the foci is located on the x axis is given as

Figure 2: Different parts of a hyperbola

Following are the main parts of a hyperbola

Center

In a conic section, center is defined as the point that bisect every chord passing through the point. For a hyperbola, center is the mid-point of the line segment that joins vertices of the hyperbola. The center of a hyperbola lies at (h, k).

Vertices

Vertices of a hyperbola refer to the points where the hyperbola makes the sharpest turn. The vertices lie on the major axis. The coordinates of vertices are given as (h + a, k) and (h – a, k).

Transverse axis  

Transverse axis the line that starts from one axis, passes through the center and ends at the other vertex is called transverse axis. In other words, the line that joins the two vertices is known as transverse axis. For a hyperbola where the foci are located on the x axis, equation of the transverse axis is y = k. The length of transverse axis is 2a unit.

Co-Vertices

Co-vertices are associated to b that is the minor semi-axis length. The coordinates of co-vertices are given as (h, k + b) and (h, k – b).

Conjugate axis

Conjugate axis refers to the line that passes through the center of the hyperbola and is perpendicular to the line passing through two foci. For a hyperbola where the foci are located on the x axis, equation of the conjugate axis is x = h. The length of the conjugate axis is 2b unit.

Asymptotes

The major and minor axes of a hyperbola ‘a’ and ‘b’ together with vertices and co-vertices form a rectangle sharing same center with the hyperbola. The dimension of the rectangle is given as 2a × 2b. Asymptotes of a hyperbola refers to straight lines forming diagonals of the rectangle. The equation of the asymptotes lines can therefore be obtained using corners of the rectangle. The equation of asymptotes lines hence is given as

The rectangle has useful application for drawing graph of the hyperbola since these lines contains vertices. While drawing the hyperbola, the rectangle has to be drawn first. Then the asymptotes are drawn as extended lines forming diagonals of the rectangle. The hyperbola then is drawn by following the asymptote inwards. The curve is drawn such a way that it touches vertex of this rectangle. The other curve is drawn as asymptotes out. The process is repeated for other branch.

Focal point

Focal points or foci of a hyperbola are the two fixed points that are located inside each curve of the hyperbola. Foci are used in the formal definition of hyperbola. A hyperbola then is defined as for two fixed points called Foci, hyperbola is the set of all points having same differences of distances to each focal points. The coordinates of two focal points of the hyperbola is given as (h + c, k) and (h – c, k). The value of c in this case is obtained using the following relation.

Application of hyperbola to real world problems

In real world, hyperbola is used in many aspects. Some of its real world application is discussed below.

Dulles Airport

The Dulles Airport as designed by Eero Saarinen resembles the shape of a hyperboloid paraboloid. This hyperboloid paraboloid has a three dimensional curve located in one cross section. There is also a parabola in other side of the cross section.

Lampshade

The shadows caste from a lampshade form hyperboloid shape on the wall.

Gear transmission

The hyperboloid shape is particularly useful in gear transmission. Two hyperboloids of revolution can give gear transmission between the two skew axes. Cogs of each of the gear consist of set of straight line.

Cooling Towers of Nuclear Reactors and Coal-fired Power Plants

Almost all nuclear cooling towers and some coal-fired plants follows the standard design of a hyperboloid shape. These are structurally sound and are built with steel beams. In building the cooling towers two kinds of problem are generally faced by engineers. First, the structure has to be capable of standing in face of high winds. Second, it should be built with using as minimal materials as possible.

A hyperboloid shape can successfully address both these problems. Given the diameter and height of a tower with a given strength, a hyperbolic shape requires less materials compared to any other shapes. For example, a 500-foot-high tower can possibly be made by using concrete shell having width of only six to eight inches.

Stones in a lake

When two stones are simultaneously thrown into a lake with still water, the ripples are moved outward in the shape of concentric circles. The circles intersect each other at a point and form a curve in the shape of a hyperbola.